%I #19 Nov 06 2016 01:48:48

%S 1,6,5,3,2,3,1,5,9,5,9,7,6,1,6,6,9,6,4,3,8,9,2,7,0,4,5,9,2,8,8,7,8,5,

%T 1,7,4,3,8,3,4,1,2,9,0,7,0,2,5,5,1,8,6,8,8,6,1,1,7,7,9,3,6,5,9,5,6,0,

%U 2,7,0,3,0,9,4,9,5,1,0,8,5,2,3,0,7,7,8

%N Decimal expansion of Sum'_{(x,y,z)=-infinity..infinity} 1/(x^2+y^2+z^2)^2, where the 'prime' indicates that the term x=y=z=0 is to be left out.

%H Jon E. Schoenfield, <a href="/A259281/b259281.txt">Table of n, a(n) for n = 2..401</a>

%H M. Kontsevich and D. Zagier, <a href="http://www.ihes.fr/~maxim/TEXTS/Periods.pdf">Periods</a>, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22 p. 25.

%H Jon E. Schoenfield, <a href="/A259281/a259281_1.txt">Magma program</a>

%H I. J. Zucker, <a href="http://dx.doi.org/10.1088/0305-4470/8/11/008">Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures</a>, J. Phys. A: Math. Gen. 8 (1975) 1734, Table 4, s=2, a(2s).

%e 16.532315959761669643892704592887851743834129...

%t (* This script gives only 10 correct digits *) s1 = NSum[(-2 + Pi*x*(Coth[Pi*x] + Pi*x*Csch[Pi*x]^2))/(4*x^4), {x, 1, Infinity} ];s2 = NSum[-((Csch[Pi*x]^2*(2 + 2*Pi^2*x^2 - 2*Cosh[2*Pi*x] + Pi*x*Sinh[2*Pi*x]))/(16*x^4)), {x, 1, Infinity} ]; f[y_?NumericQ] := NSum[(Pi*Coth[Pi*Sqrt[x^2 + y^2]])/(4*(x^2 + y^2)^(3/2)), {x, 1, Infinity} ]; s3 = NSum[f[y], {y, 1, Infinity} ]; g[y_?NumericQ] := NSum[2*((Pi^2*x^2*Csch[Pi*Sqrt[x^2 + y^2]]^2)/(4*(x^2 + y^2)^2)), {x, 1, Infinity} ]; s4 = NSum[g[y], {y, 1, Infinity} ]; s = Pi^4/15 + 12*s1 + 8*(s2 + s3 + s4); RealDigits[s, 10, 10] // First

%K nonn,cons

%O 2,2

%A _Jean-François Alcover_, Jun 23 2015